A crystalline measure is an atomic measure $\mu$ which is supported by a locally finite set $\Lambda$ and whose distributional Fourier transform $\widehat \mu$ is also supported by a locally finite set $F$. A generalized Poisson summation formula is associated with a crystalline measure. The simplest example is given by the usual Poisson summation formula and the Dirac comb. Crystalline measures were introduced by Andrew Guinand in 1958. Riemann used the usual Poisson summation formula to prove the functional equation satisfied by the zeta function. This will be extended to any crystalline measure. Next, we give a new proof of a spectacular theorem by Pavel Kurasov and Peter Sarnak. They constructed infinitely many crystalline measures of the form $\mu=\sum_{\lambda\in \Lambda}\delta_\lambda$ where $\Lambda$ is a uniformly discrete set. Our approach is based on the theory of $\it{curved}$ $\it{model}$ $\it{sets}$. Then crystalline measures will be related to some interpolation problems in the time-frequency plane. This leads to the solution given by Maryna Viazovska of Kepler's problem in dimension 8.